Search for $e^+ e^- \to \gamma\chi_{bJ}$ ($J$ = 0, 1, 2) near $\sqrt{s} = 10.746$ GeV at Belle II

Using Belle II data collected at SuperKEKB, this study searches for the $e^+ e^- \to \gamma\chi_{bJ}$ ($J$ = 0, 1, 2) processes near $\sqrt{s} = 10.746$ GeV and establishes 90% confidence level upper limits on their Born cross sections, finding them significantly smaller than those of related processes like $e^+e^-\to\omega\chi_{b1}$.

The Gist

Imagine the universe as a giant, high-speed racetrack. In this race, tiny particles called electrons and positrons (the antimatter twins of electrons) zoom toward each other at nearly the speed of light. When they crash, they don't just break; they explode into a shower of new, exotic particles. This is what happens at the SuperKEKB collider in Japan, where the Belle II detector acts like a massive, ultra-high-speed camera trying to capture every detail of the crash.

The Mystery: The "Ghost" Particle

Scientists have been studying a specific "ghost" particle called $\Upsilon(10753)$. Think of this particle as a mysterious celebrity that appeared on the racetrack a few years ago. We know it exists, but we don't know what it is made of. Is it a simple family of particles (a "conventional" bottomonium)? Is it a weird hybrid? Or is it a "tetraquark" (a particle made of four quarks glued together)?

To solve this mystery, physicists need to see how this celebrity behaves when it decays (breaks apart). They are looking for specific "footprints" left behind.

The Search: Looking for a Photon and a "Charm"

In this specific study, the scientists were hunting for a very specific reaction:
Electron + Positron $\rightarrow$ Photon ($\gamma$) + $\chi_{bJ}$

Let's break this down with an analogy:

• The Crash: Two cars (electron and positron) smash together.
• The Photon ($\gamma$): A flash of light (like a camera flash) shooting out.
• The $\chi_{bJ}$: A heavy, heavy suitcase that immediately falls apart into a photon and a bottomonium particle (specifically the $\Upsilon(1S)$).
• The Final Clue: The bottomonium particle then falls apart into a pair of lighter particles (either two electrons or two muons).

So, the scientists are looking for a final scene with two photons and two heavy particles (electrons or muons) flying out of the crash.

The Detective Work

The Belle II team took data at four different "speeds" (energy levels) near 10.746 GeV. They had a massive amount of data (equivalent to billions of collisions).

1. Filtering the Noise: Most crashes are boring. They produce standard particles. The scientists had to build a digital filter to ignore the "noise" (like standard Bhabha scattering, which is just electrons bouncing off each other) and focus only on the rare events that looked like their target.
2. The Kinematic Fit: Imagine trying to solve a puzzle where you know the total weight of the pieces before the crash. The scientists used math to check if the pieces they found (the photons and electrons) added up perfectly to the energy of the original crash. If the math didn't add up, it wasn't the right particle.
3. The Search: They looked at the "invariant mass" (a way of calculating the weight of the combined particles) to see if there was a "bump" in the data. A bump would mean a new particle was found.

The Result: The Ghost Remains Elusive

After analyzing millions of collisions, the scientists found nothing.

• No Signal: There was no "bump" in the data. The graph looked like a flat, empty field.
• The Conclusion: The process $e^+e^- \rightarrow \gamma\chi_{bJ}$ does not happen often enough to be seen with the current amount of data.

However, "nothing" is still a result! The scientists set Upper Limits.

• The Analogy: Imagine you are looking for a specific type of rare bird in a forest. You spend 100 hours looking and see zero birds. You can't say the bird doesn't exist, but you can say: "If this bird exists, it is so rare that it appears less than once every 100 hours."
• The paper says: "If this reaction happens, it is extremely rare—much rarer than other similar reactions we have seen before."

Why Does This Matter?

Even though they didn't find the particle, this result is a huge clue for theorists.

• The Theory: Some theories predicted that if the mysterious $\Upsilon(10753)$ is a specific type of particle (a "2D state"), it should decay into a photon and a $\chi_{bJ}$ very frequently.
• The Reality: Since the scientists didn't see it, those specific theories are now under pressure. It suggests the $\Upsilon(10753)$ might not be that specific type of particle after all.

Summary

The Belle II collaboration acted like cosmic detectives, scanning billions of high-speed crashes for a specific, rare pattern of light and matter. They didn't find the "smoking gun" (the particle), but they successfully proved that if the gun exists, it's hiding very well. This helps scientists narrow down the list of suspects to figure out exactly what the mysterious $\Upsilon(10753)$ really is.

Technical Summary

1. Problem and Motivation

The paper addresses the nature of the $\Upsilon(10753)$ resonance, first observed by the Belle collaboration in 2019. This state, with a mass of approximately 10.753 GeV/$c^2$, exhibits a complex structure in the $e^+e^- \to \Upsilon(nS)\pi^+\pi^-$ cross-section, showing two closely spaced peaks ($\Upsilon(10753)$ and $\Upsilon(10860)$). Theoretical interpretations of $\Upsilon(10753)$ vary widely, including:

• A conventional bottomonium state (specifically a $2D$ state).
• A hybrid meson.
• A tetraquark state.

To distinguish between these models, studying the decay modes of $\Upsilon(10753)$ is crucial. While hadronic transitions like $\Upsilon(10753) \to \omega\chi_{bJ}$ have been observed, radiative transitions ($\Upsilon(10753) \to \gamma\chi_{bJ}$) had not been studied experimentally. Theoretical calculations suggest that if $\Upsilon(10753)$ is a pure $2D$ state, the branching fractions for $\gamma\chi_{b1}$ and $\gamma\chi_{b2}$ should be significant ($>10^{-2}$). Furthermore, analogies with charmonium-like states (e.g., $Y(4230)$ decaying to $\gamma\chi_{cJ}$) suggest similar dynamics might exist in the bottomonium sector.

2. Methodology

Data Sample

The analysis utilizes data collected by the Belle II detector at the SuperKEKB collider. The dataset corresponds to four center-of-mass energy ($\sqrt{s}$) points near the $\Upsilon(10753)$ resonance:

• 10.653 GeV: 3.5 fb$^{-1}$
• 10.701 GeV: 1.6 fb$^{-1}$
• 10.746 GeV: 9.8 fb$^{-1}$ (Peak energy)
• 10.804 GeV: 4.7 fb$^{-1}$
• Total Integrated Luminosity: ~19.6 fb$^{-1}$

Event Reconstruction and Selection

The search targets the process $e^+e^- \to \gamma\chi_{bJ}$, where $\chi_{bJ}$ decays to $\gamma\Upsilon(1S)$, and $\Upsilon(1S)$ decays to $e^+e^-$ or $\mu^+\mu^-$. The final state is therefore $\gamma\gamma\ell^+\ell^-$ ($\ell = e, \mu$).

• Track Selection: Events must contain exactly two charged tracks. Tracks must originate close to the interaction point (transverse $< 1$ cm, longitudinal $< 3$ cm).
• Particle Identification (PID): Leptons are identified using likelihood ratios ($R_e > 0.9$ or $R_\mu > 0.9$), achieving ~95% and ~90% efficiency respectively.
• Photon Selection: Two photons are required.
  • $\gamma_h$ (High energy): Assumed to come from the primary interaction.
  • $\gamma_l$ (Low energy): Assumed to come from the $\chi_{bJ}$ decay.
  • Energy thresholds: $\gamma_l > 280$ MeV ($\mu^+\mu^-$ mode) and $> 290$ MeV ($e^+e^-$ mode).
  • Shower shape requirements (3$\times$3 vs 5$\times$5 crystal matrix ratio $> 0.8$).
  • Angular cuts on $\gamma_h$ to suppress Bhabha scattering.
• Kinematic Fit: A kinematic fit constrains the four-momentum and vertex to the initial $e^+e^-$ collision and interaction region. The dilepton invariant mass ($M_{\ell\ell}$) is constrained to the $\Upsilon(1S)$ signal region ($9.44 < M_{\ell\ell} < 9.49$ GeV/$c^2$). Candidates with $\chi^2 < 30$ are retained.

Statistical Analysis

• Fitting: Unbinned extended maximum-likelihood fits are performed on the $\gamma\Upsilon(1S)$ invariant mass distribution ($M_{\gamma\Upsilon(1S)}$).
• Signal Model: A sum of a Crystal Ball function and a Gaussian function (sharing a common mean fixed to known $\chi_{bJ}$ masses).
• Background Model: A second-order Chebyshev polynomial.
• Simultaneous Fit: $\mu^+\mu^-$ and $e^+e^-$ modes are fitted simultaneously, with yield ratios fixed by branching fractions and efficiencies.
• Upper Limits: Since no significant signal was observed, 90% Confidence Level (C.L.) upper limits on signal yields ($N_{UL}$) and Born cross sections ($\sigma_{Born}$) were calculated using profile likelihoods.

Systematic Uncertainties

Major sources include:

• Branching fractions of intermediate states (dominant: 7.2–14.6%).
• Reconstruction efficiency (lepton and photon).
• Trigger simulation.
• Integrated luminosity (1.2%).
• Beam energy calibration and radiative corrections.
• Fit model variations and mass/width uncertainties.

3. Key Contributions

• First Experimental Search: This is the first experimental study of the radiative transitions $e^+e^- \to \gamma\chi_{bJ}$ ($J=0,1,2$) in the energy region of the $\Upsilon(10753)$.
• Comprehensive Energy Scan: The analysis covers four energy points, allowing for a detailed scan of the cross-section behavior near the resonance peak.
• Rigorous Limits: The paper establishes the most stringent upper limits to date on the Born cross sections for these processes.

4. Results

• No Significant Signal: No evidence for $e^+e^- \to \gamma\chi_{bJ}$ was found at any of the four energy points. The largest statistical significance observed was 2.3$\sigma$ for $\gamma\chi_{b0}$ at $\sqrt{s} = 10.804$ GeV, which is below the discovery threshold.
• Upper Limits on Cross Sections:
  • At the peak energy $\sqrt{s} = 10.746$ GeV, the 90% C.L. upper limits on the Born cross sections are:
    • $\sigma(e^+e^- \to \gamma\chi_{b0}) < 5.37$ pb
    • $\sigma(e^+e^- \to \gamma\chi_{b1}) < 0.26$ pb
    • $\sigma(e^+e^- \to \gamma\chi_{b2}) < 0.79$ pb
  • The limit for $\chi_{b1}$ is particularly tight.
• Comparison with Other Modes: The upper limits for $\gamma\chi_{b1}$ are significantly smaller than the measured cross sections for hadronic transitions like $e^+e^- \to \omega\chi_{b1}$ and $e^+e^- \to \pi^+\pi^-\Upsilon(2S)$ at the same energy. This suggests that radiative transitions are suppressed compared to hadronic transitions for this resonance.

5. Significance

• Constraints on Theoretical Models: The non-observation of these radiative decays, combined with the small upper limits, provides critical constraints on theoretical models of the $\Upsilon(10753)$.
  • If $\Upsilon(10753)$ were a pure $2D$ bottomonium state, large branching fractions for $\gamma\chi_{b1/2}$ were predicted. The small upper limits challenge this pure assignment or suggest significant mixing with other configurations (e.g., tetraquark or hybrid components).
• Insight into Dynamics: The results highlight a difference in dynamics between the bottomonium and charmonium sectors (where BESIII observed $\gamma\chi_{cJ}$ signals for $Y(4230)$). The suppression of radiative decays in $\Upsilon(10753)$ suggests different internal structures or decay mechanisms compared to its charmonium analogs.
• Future Input: These measurements, when combined with existing hadronic transition data, provide essential input for clarifying the underlying nature of the $\Upsilon(10753)$ and the broader spectrum of exotic heavy quarkonium states.